Post-1-Newtonian quadrupole tidal interactions in binary systems

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by  Justin Vines

Post-1-Newtonian quadrupole tidal interactions in binary systems
Justin E. Vines and E´anna E´. Flanagan
Department of Physics, Cornell University, Ithaca, NY 14853, USA.
(Dated: January 11, 2011)
Weconsiderorbital-tidalcouplinginabinarystellarsystemtopost-1-Newtonianorder. Wederive
theorbital equationsofmotion forbodieswithspinsandmassquadrupolemomentsandshowthat
they conserve the total linear momentum of the binary. Momentum conservation also allows us
to specialize ouranalysis to thesystem’s center-of-mass-energy frame; we find the binary’s relative
equation of motion in this frame and also present a generalized Lagrangian from which it can be
derived. Wethenspecializetothecaseinwhichthequadrupolemomentisadiabaticallyinducedby
the tidal field. We show how the adiabatic dynamics for the quadrupole can be incorporated into
ouractionprincipleandpresentthesimplifiedorbitalequationsofmotionandconservedenergyfor
theadiabatic case. These results arerelevant to thegravitational wavesignal of inspiralling binary
1 neutron stars.
1
0
2 I. INTRODUCTION AND SUMMARY λ to 10%, with the largest source of error being post-
∼
1-Newtonian (1PN) corrections to the tidal-orbital cou-
n
a A. Background and motivation pling [8, 9]. It is our goal in this paper to lay the foun-
J dations for computing the tidal signal to 1PN accuracy.

9 Inspiralling and coalescing compact binaries present The modeling of tidal effects in GW signals from neu-
tronstarbinariescanbedividedintothreeseparateprob-
oneofthemostpromisingsourcesforground-basedgrav-
] lems: (i) to calculate the deformation response of each
c itationalwave(GW)detectors[1]. Aprimarygoalinthe
q measurement of GW signals from neutron star-neutron neutronstartothetidalfieldgeneratedbyitscompanion,
- star (NSNS) and black hole-neutron star (BHNS) bina- (ii)tocalculatetheinfluenceofthetidaldeformationson
r
the system’s orbital dynamics, and (iii) to calculate the
g riesis toprobethe neutronstarmatter equationofstate
[ (EoS),whichiscurrentlyonlylooselyconstrainedbyelec- gravitationalwaveformemittedbythesystemandincor-
tromagnetic observations in the relevant denisty range poratethecorrespondingradiationreactioneffectsinthe
2
v ρ 2–8 1014 g cm3 [2]. The EoS will leave its imprint orbitaldynamics. This paper is focused onsolvingprob-
∼ × lem (ii) to 1PN accuracy, while we leave problem (iii) to
9 onthe GW signalvia the effects of tidal coupling,as the
1 neutron star is distorted by the non-uniform field of its future work. A solution to problem (i) that is sufficient
9 companion. Many recent studies into the effects of the for our purposes has been given in Refs. [9–12].
4 neutronstar EoSon binaryGW signals havebeen based Tidal Deformability Computations: While computing
.
9 on numerical simulations of the fully relativistic hydro- tidal deformations of stars is a well studied problem in
0 dynamical evolution of NSNS and BHNS binaries (see Newtoniangravity[13],it hasrecentlybeenre-examined
0 e.g.thereviews[3,4]). Thesesimulationshavelargelyfo- in the context of fully relativistic stars by Hinderer et
1
cusedonthe binaries’lastfeworbitsandmerger,atGW al. [9, 10], Damour and Nagar [11], and Binnington and
:
v frequencies &500 Hz, and have investigated constraining Poisson [12]. These authors used fully relativistic mod-
i neutron star structure via (for example) the GW energy els for the star’s interior, with various candidate equa-
X
spectrum [5], effective cutoff frequencies at merger [6], tionsofstate,tocalculatethe perturbationstothestar’s
r
a and tidal disruption signals [7]. equilibrium configuration produced by a static external
AsrecentlyinvestigatedbyFlanaganandHinderer[8], tidal field. In Refs. [9, 10], the results were used to de-
neutron star internal structure may also have a measur- terminethe(electric-type)quadrupolartidaldeformabil-
able influence on the GW signalfrom the earlier inspiral ity λ, while Refs. [11, 12] extended the analysis and in-
stage of a binary’s orbital evolution, at GW frequencies cludedallhigher-multipolarelectric-typeresponsecoeffi-
.500Hz. While tidal coupling will produce only a small cients (for the mass multipole moments) as well as their
perturbationtothe GWsignalinthislow-frequencyadi- magnetic-type analogues (for the current multipole mo-
abatic regime,the tidal signalshould be relativelyclean, ments). While these results concern a star’s response to
depending (at leading order) on a single parameter per- astatictidalfield,theyarestillapplicableforinspiralling
taining to the neutron star structure. This tidal de- binaries atsufficiently low orbitalfrequencies; a star will
formability parameter λ is the proportionality constant approximately maintain equilibrium with the instanta-
between the applied tidal field and the star’s induced neous tidal field if the field changes adiabatically.
quadrupole moment and is sensitive to the neutron star Orbital Dynamics including Quadrupole: In their
EoS. The measurement scheme proposed in Ref. [8] is treatmentoftidaleffectsinbinaryGWsignalsinRef.[8],
based on an analytical model for the tidal contribution Flangan and Hinderer used Newtonian gravity to treat
to the GW signal, giving a linear perturbation to the (the conservativepartof)the system’s orbitaldynamics;
GW phase proportional to λ. At GW frequencies .400 they estimated that 1PN corrections to the orbits would
Hz,themodelshouldbesufficientlyaccuratetoconstrain modify the calculated tidal signal by 10%. A general
∼
2
formalism for calculating 1PN corrections to the orbital canbecharacterizedbyderivativesofanexternalNewto-
dynamics ofextended bodies hasbeen developedina se- nianpotentialφ (t,x); inabinarysystem,theleading-
ext
ries of papers [14, 15] by Damour, Soffel, and Xu (DSX) order potential is φ = GM/r, where G is Newton’s
ext
−
and later extended by Racine and Flanagan[16]. In this constant, M is the mass of the companion, and r the
paper,weapplythatformalismtodetermineandanalyze distance between the body and its companion.
theexplicit1PNtranslationalequationsofmotionforbi- Thenon-uniformfieldofthecompanionproducestidal
nary systems with quadrupolar tidal interactions. While forcesonthenon-sphericalbody(inadditiontotheusual
suchequationsofmotionhavepreviouslybeenpresented 1/r2 force) according to
in Refs. [17, 18], we find that those results differ from
ours. We also extend previous results by analyzing 1PN 1 1
Mz¨i = M∂ + Qjk∂ + Qjkl∂ +... φ
momentum conservation for the binary system (which − i 2 ijk 6 ijkl ext
(cid:18) (cid:19)
serves as a strong consistency check for our equations M Qij Qijk
of motion, and one not satisfied by the results [17, 18]), GM + | | + | | +...
∼ r2 r4 r5
by specializing the equations of motion to the system’s (cid:20) (cid:21)
center-of-mass-energy frame, and by formulating an ac- GM2 R2 R3
1+ +O , (1.2)
tion principle for the orbital dynamics. ∼ r2 r2 r3
(cid:20) (cid:18) (cid:19)(cid:21)
Gravitational Wave Emission: To calculate the emit-
ted GW signal and radiation reaction effects, Flanagan where M is the mass of the body or its companion, as-
and Hinderer [8] used the quadrupole radiation approx- sumed here to be roughly equal, and the derivatives of
imation and associated 2.5PN radiation reaction forces. the external potential are evaluated at the body’s center
To consistently generalize their calculation in Ref. [8] to of mass, xi = zi(t). The contributions to the net force
(relative)1PNorder,it willbe necessaryto compute the fromthequadrupoleandhigher-ordermultipolesarethus
3.5PN corrections to the GW generation, in addition to seen to take the form of an expansion in R/r, the ratio
the 1PN orbital corrections considered in this paper; we of the size of the body to the orbital separation. When
leave these calculations to future work. this finite-size parameter is small, as in the early stages
of binary inspiral, the tidal force is well approximated
Theissueoftidaleffectsininspiral-stageNSNSbinary
bythequadrupolartermalone. Amoredetailedaccount
GW signals has also been recently addressed by means
of Newtonian tidal forces and related results is given in
of numerical relativity simulations in Refs. [19, 20], and
Sec. II below.
analytically,within the context of the effective one body
In neutron star binaries, a quadrupole is induced by
(EOB) formalism, by Damour and Nagar [21]. Ref. [21]
differential forces resulting from the non-uniformfield of
proposes a contribution to the EOB “radial potential”
thecompanion. Intheadiabaticlimit,whentheresponse
that should accountfor the same 1PNcorrections to the
time scale of the body is much less than the time scale
quadrupole-tidalinteractionconsideredin this paper. In
onwhich the tidalfield changes,the induced quadrupole
AppendixA,wedemonstratetheequivalenceoftheEOB
will be given (to linear order in the tidal field) by
Hamiltonian of Ref. [21] and our Lagrangian (1.12) for
circular orbits; we also show how the EOB Hamiltonian
Qij(t)= λ∂ ∂ φ (t,z), (1.3)
can be extended to include 1PN tidal effects for non- − i j ext
circular orbits.
withtheconstantλbeingthetidaldeformability. Thisis
related to the more often used dimensionless Love num-
ber k by λ = 2k R5/3G, where R is the body’s radius
2 2
B. Newtonian tidal coupling [13].
Using Newtonian gravity to describe the orbital dy-
Gravitational tidal coupling arises from the interac- namics and the adiabatic approximation to model the
tion of the non-spherical components of a body’s matter stars’ induced quadrupoles, Flanagan and Hinderer [8]
distribution with a non-uniform gravitational field. The calculated the effect of tidal interactions on the phase
non-sphericity is characterized at leading order by the of the gravitational waveform emitted by an inspiralling
body’s mass quadrupole moment, neutronstarbinaryandanalyzedthemeasurabilityofthe
tidaleffects. TheyfoundthatAdvancedLIGOshouldbe
1 able to constrain the neutron stars’ tidal deformability
Qij(t)= d3xρ(x¯ix¯j δij x¯ 2), (1.1)
− 3 | | to λ (2.0 1037gcm2s2)(D/50Mpc) with 90% con-
Z ≤ ×
fidence, for a binary of two 1.4 M neutron stars at a
⊙
withρ(t,x)beingthemassdensityandx¯i(t)=xi zi(t) distance D from the detector, using only the portion of
−
being the displacement from the body’s center of mass the signal with GW frequencies less than 400 Hz. The
position xi = zi(t). The quadrupole is (at most) on the calculationsofλfora1.4M neutronstarinRefs.[9–12],
⊙
orderofQij MR2, with M being the body’s mass and using several different equations of state, give values in
R its radius.∼ Higher-order deformations are described the range 0.03–1.0 1037gcm2s2, so that nearby events
by the octupole, Qijk d3xρx¯ix¯jx¯k MR3, and mayallowAdvance×dLIGOtoplaceusefulconstraintson
∼ ∼
higher-order multipole moments. The non-uniform field candidate equations of state.
R
3
Refs. [8, 9] estimate the fractional corrections to surpressedbyhigherpowersofR/randwhichweneglect
the tidal signal due to several effects neglected by the in our analysis [16].
model of the GW phasing used in Ref. [8], namely, The first term in the bottom line represents the 1PN
non-adiabaticity (.1%), higher-multipolar tidal cou- spin-orbit coupling [15]. Though it would be convenient
pling (.0.7%),nonlinear hydrodynamic effects (.0.1%), to be able to specialize to the case of non-spinning bod-
spin effects (.0.3%), nonlinear response to the tidal ies when studying tidal interactions, tidal fields will in
field (.3%), viscous dissipation (negligible), and post- general lead to tidal torques on non-spherical bodies; a
Newtonianeffects(.10%). Thelargestcorrections,from consistent 1PN treatment of tidal coupling thus requires
post-Newtonian effects in the orbital dynamics and GW the inclusion of 1PN spin-orbit coupling terms. The fi-
emission, will depend on the neutron star physics only nal term of the bottom line denotes 1PN contributions
through the same tidal deformability parameter λ used to the EoMs from the bodies’ current quadrupoles and
in the Newtonian treatment and thus can be easily in- higher current multipoles, which we will not include in
corporated into the data anaysis methods outlined in our analysis.
Refs. [8, 9]. TheDSXtreatmentof1PNcelestialmechanics[14,15]
providesaframeworkforcalculatingtheorbitalEoMsfor
bodieswitharbitrarilyhigh-ordermassandcurrentmul-
C. Post-1-Newtonian corrections tipole moments. In Ref. [15], DSX applied their formal-
ism to rederive the explicit 1PN EoMs for bodies with
mass monopoles and current dipoles (spins). The calcu-
Forinspriallingneutronstarbinarieswithatotalmass
lationwasthenextendedtoincludetheeffectsofthebod-
of 3M atorbitalfrequenciesof 200Hz(GWfrequen-
⊙
∼ ∼ ies’ mass quadrupoles by Xu et al [17, 18]. Racine and
cies of 400 Hz), the post-Newtonian expansion param-
∼ Flanagan (RF) [16] later reworked the DSX formalism
eter v2/c2 GM/c2r is 0.1, so that the 1PN approxi-
∼ ∼ and presented explicit 1PN EoMs for bodies with arbi-
mationiswellsuitedtodescribingrelativisticcorrections
trarilyhigh-ordermultipoles,whichcanbespecializedto
to the binary orbit. As discussed in depth in Sec. III be-
thecaseofbodieswithonlyspinsandmassquadrupoles.
low, the 1PN orbital dynamics of a binary system with
Wehavefound,however,thatthefinalresultsofXuetal
tidal coupling can be described by translational equa-
andthoseofRFareindisagreementwitheachotherand
tions of motion (EoMs) similar in form to the Newto-
with our recent calculations. Some typos and omissions
nian equations schematically represented in (1.2), giving
leading to errorsin RF have been identified and are out-
the center-of-mass acceleration of each constituent body
lined in an upcoming erratum; the results given in this
in terms of their positions and multipole moments. The
paper agree with the corrected results of RF. In Sec. III
1PNequationsofmotionaddorder1/c2correctionterms
below, we review the essential ideas of the DSX formal-
to (1.2) which depend not only on the bodies’ positions
ism, following the notations and conventions of RF, and
and mass multipole moments, M, Qij, Qijk, etc., but
weoutlinethefullprocedurebywhichourresultsforthe
also on their velocities and current multipole moments,
1PN EoMs are derived.
Si, Sij, etc. Expanding in both the post-Newtonian pa-
rameterv2/c2andthefinitesizeparameterR/r,the1PN
equations of motion can be written schematically as
D. Summary of results
M v2M v4M
Mz¨i GM + +O
∼ r2 c2 r2 c4 r2 1. The M1-M2-S2-Q2 system
(cid:20) (cid:18) (cid:19)
Qij v2 Qij Qijk R3M
+| | + | | +O | | Ourresultsconcernthe1PNgravitationalinteractions
r4 c2 r4 r5 ∼ r3 r2
(cid:18) (cid:19) in a system of two bodies, which we label “1” and “2”.
v Si v Sij v2R2M We model body 1 as an effective point particle, with
+ | | +O | | (1.4)
c2r3 c2r4 ∼ c2 r2 r2 a mass monopole moment only, while we take body 2
(cid:18) (cid:19)(cid:21)
to have additionally a spin and a mass quadrupole mo-
In the top line, the first term gives the usual point- ment. We consistently work to linear order in the spin
particle (monopole) force of Newtonian gravity, and the andquadrupole,andourresultscanthusbeeasilygener-
second term represents its 1PN corrections. The last alized to the case of two spinning, deformable bodies by
term of the top line denotes 2PN and higher post- interchanging particle labels. We initially assume noth-
Newtonian order corrections, which we neglect in this ingaboutthebodies’internalstructureordynamics. Our
paper. Point-particle EoMs are in fact currently known primaryassumptionisthevalidityofthe1PNapproxima-
up through post-3-Newtonian order [22]. tiontogeneralrelativityinavacuumregionsurrounding
In the second line of Eq. (1.4), we have first the New- the bodies.
tonian quadrupole-tidal term, followed by its 1PN cor- The system’s orbital dynamics can be formulated
rections(whicharethesubjectofthispaper),andfinally in terms of the bodies’ center-of-mass worldlines zi(t)
1
contributions from the octupole and higher mass multi- and zi(t) and their multipole moments: the mass
2
poles (and their post-Newtonian corrections) which are monopolesM (t)andM (t),thespinSi(t),andthemass
1 2 2
4
quadrupole Qij(t). (c.f. Sec. IIF). Eqs. (1.6) ensure that M satisfies the
2 2
The wordlines xi = zi(t) and xi = zi(t) parametrize 1PN evolution equation (3.26,3.27). The decomposition
1 2
the bodies’ positions to 1PN accruacy in a (confor- of the monopole M in (1.6a) is essential for properly
2
mally Cartesian and harmonic) ’global’ coordinate sys- formulating an action principle for the orbital dynam-
tem (t,xi), which tends to an inertial coordinate system ics. The evolution of the spin Si is determined by the
in Minkowski spacetime as x . The global coor- (Newtonian-order) tidal torque formula:
| | → ∞
dinates and center-of-mass worldlines are defined more
precisely in Sec. IIID. We use the following notation for S˙i = 3M1ǫijkQjanank+O(c−2), (1.7)
the relative position and velocity: r3
zi =zi zi, r = z = δ zizj, ni =zi/r,
2− 1 | | ij as in (2.45). Finally, the 1PN translational equations of
vi =z˙i, vi =z˙i, vi =vi vi, r˙ =vini, motion, which govern the evolution of the worldlines zi
1 1 2 2 p 2− 1 1
and zi, are of the form
with dots denoting derivativeswith respectto the global 2
frame time coordinate t.
The multipole moments—M1(t), M2(t), S2i(t), and z¨1i = F1i(zj,v1j,v2j,M1,M2,Sj,Qjk,Q˙jk,Q¨jk),
Qi2j(t) for our truncated system—are defined in z¨2i = F2i(zj,v1j,v2j,M1,M2,Sj,Qjk,Q˙jk),
Secs.IIIBandIIIEviaamultipoleexpansionofthe1PN
metricinavacuumregionsurroundingeachbody[16]. In and are given explicitly by Eqs. (3.31).
thecaseofweaklyself-gravitatingbodies,thesemoments In Sec. IV, we define and calculate the 1PN-accurate
can be defined as intergrals of the stress-energy tensor mass dipole moment of the entire system Mi (t), given
sys
overthevolumeofthebodies,asinEqs.(3.12);thesedef- in Eq. (4.8). We find that the condition M¨i =O(c−4),
sys
initionscoincidewiththoseoftheBlanchet-Damourmul- required by Einstein’s equations and reflecting the con-
tipole moments introducedinRef. [23]andusedby DSX servation of the system’s total momentum, is satisfied
[14,15]. Themassmultipolemoments(likeM1,M2,and as a consequence of the orbital EoMs (3.31); this serves
Qij) are defined with 1PN accuracy, while the current as a non-trivial consistency check for our results. The
2
multipole moments (like Si) appear only in 1PN-order conservation of momentum also allows us to specialize
2
terms andthus needonly be definedwith Newtonianac- theEoMstothesystem’scenter-of-mass(-energy)(CoM)
curacy. We will often denote the spin and quadrupole of frame, which can be defined by the condition Mi = 0
sys
body 2 by S and Qij, dropping the “2” labels. as in Sec. VA. The two EoMs (3.31) for the worldlines
i
zi and zi can then be traded for the single EoM for
1 2
the CoM-frame relative position zi = zi zi, given in
2. General equations of motion and orbital Lagrangian Eq. (5.9). 2 − 1
Ourresultscanbemostcompactlysummarizedbygiv-
The equations of motion for the monopoles M (t) and ing a Lagrangian formulation of the CoM-frame orbital
1
M (t), the spin Si(t), and the worldlines zi(t) and zi(t) dynamics, as discussed in Sec. (VB). We find that the
2 1 2
are determined by Einstein’s equations alone, while that CoM-frame orbital EoM (5.9) can be derived from the
for the quadrupole Qij(t) will depend on the details of Euler-Lagrangeequation
body 2’s internal dynamics and can initially be left un-
specified. The mass monopole of body 1, the effective ∂ d ∂ d2 ∂
+ =0, (1.8)
point particle, is found to be conserved to 1PN order: ∂zi − dt∂vi dt2∂ai Lorb
(cid:18) (cid:19)
M˙ =O(c−4), (1.5)
1 with a generalized Lagrangian given by
orb
L
while that of body 2 is not. As discussed in Sec. IIIF,
the1PN-accuratemassmonopoleM canbedecomposed = + + , (1.9a)
2 orb M S Q
L L L L
according to
with the monopole part,
M = nM +c−2 Eint+3U +O(c−4). (1.6a)
2 2 2 Q
Here, nM2 is the New(cid:0)tonian-order(cid:1)(rest mass) contribu- = µv2 + µM + µ 1−3ηv4 (1.9b)
tion, which is conserved: LM 2 r c2 8
(cid:20)
nM˙ =0. (1.6b) M M
2 + ηr˙2+(3+η)v2 +O(c−4),
The 1PN contributions to (1.6a) involve the Newtonian 2r − r
(cid:18) (cid:19)(cid:21)
potential energy of the quadrupole-tidal interaction,
the spin part,
3M
U = 1Qijninj, (1.6c)
Q − 2r3 χ 2M χ
andtheNewtonianinternalenergyofbody2,E2int,whose LS = c21ǫabcSavb r2 nc+ 21ac +O(c−4), (1.9c)
evolution is governedby the rate at which the tidal field (cid:20) (cid:21)
does work on the body [24]:
and the quadrupole part,
3M
E˙int = AQ˙ijninj, (1.6d)
2 2r3
5
3M 1 M M
= 1Qabnanb+ Qab nanb A v2+A r˙2+A +A vavb+A r˙navb
LQ 2r3 c2(r3 (cid:20) (cid:18) 1 2 3 r (cid:19) 4 5 (cid:21)
M M
+ Q˙ab A navb+A r˙nanb +Eint A v2+A +O(c−4). (1.9d)
r2 6 7 2 8 9 r
(cid:20) (cid:21))
(cid:2) (cid:3)
We use here the notation ization of the derivatives of the Newtonian potential in
Eq. (1.3), and λ is the 1PN-accurate tidal deformability.
M =M1+ nM2, χ1 =M1/M, χ2 = nM2/M, With the quadrupole given by (1.10), the tidal torque
µ=M nM /M, η =χ χ =µ/M, (1.7) vanishes (c.f. Eq. (2.46)), so that the spin Si is
1 2 1 2
constant; thus, in the adiabatic limit, we can specialize
withM beingthetotal(conserved)Newtonianrestmass, tothe caseSi =0withoutgeneratinginconsistencies. In
µthereducedmass,andηthesymmetricmassratio. The Sec. VIA, we show that the adiabatic evolution for the
dimensionless coefficients A –A appearing in (1.9d) are quadrupole(1.10)canbederivedfromaLagrangianthat
1 9
functions only of the mass ratios χ and χ given by adds to the orbital Lagrangianan internal elastic poten-
1 2
(5.10e). tial energy term which is quadratic in the 1PN-accurate
quadrupole:
3. Adiabatic approximation for the induced quadrupole
1
The above results concerning the orbital EoM and its = orb[zi,Qij] QabQab+O(c−4), (1.11)
L L − 4λ
Lagrangianformulation are valid regardless of the inter-
nal structure of body 2, i.e. for arbitrary evolution of its
quadrupole Qij(t). In Sec. VI, we discuss a simple adi-
abatic model for the evolution of Qij. In the adiabatic with orb given by Eq. (1.9) with Si = 0, and with
limit,thequadrupolerespondstotheinstantaneoustidal Eint =L (1/4λ)QabQab + O(c−2). (Note that any addi-
2
field according to tional constant contribution to the internal energy Eint
2
can be included as a 1PN contribution to the constant
Qij(t)=λGij(t), (1.10) nM in Eq. 1.6a.) Substituting the solution (1.10) for
2 2
Qij(t) into this Lagrangian, we obtain a simplified La-
where Gij(t), given by Eq. (6.6b), is the quadrupolar grangianinvolving only the CoM-frame relative position
2
gravito-electric tidal moment of body 2, a 1PN general- zi(t):
µv2 µM Λ µ M Λ Λ M Λ
[zi]= + 1+ + θ v4+ v2 θ +ξ +r˙2 θ +ξ + θ +ξ (1.12)
L 2 r r5 c2 0 r 1 1r5 2 2r5 r 3 3r5
(cid:18) (cid:19) (cid:26) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:27)
with Λ=(3χ /2χ )λ, and with the dimensionless coeffi- The orbital EoM resulting from this Lagranigan, which
1 2
cients can also be found by substituting the adiabatic solu-
tion (1.10) for Qij(t) directly into the general orbital
θ0 = (1 3η)/8, EoM (5.9), is given by Eq. (6.10). The conserved en-
−
θ = (3+η)/2, ergy E(z,v) derivedfrom the Lagrangian(1.12) is given
1
by Eq. (6.13). In the case of circular orbits, we find the
θ = η/2,
2
gauge-invariantenergy-frequency relationship
θ = 1/2,
3
−
ξ = (χ /2)(5+χ ),
1 1 2
ξ = 3(1 6χ +χ2),
2 − − 2 2
ξ = 7+5χ . (1.13)
3 2
−
6
1 9χ λω10/3 (9+η)(Mω)2/3 11χ λω4
E(ω)=µ(Mω)2/3 + 1 + + 1(3+2χ +3χ2) . (1.14)
−2 2χ M5/3 24 c2 4χ 2 2 Mc2
(cid:20) 2 2 (cid:21)
This result, along with others from this paper, will be bodies. We consider in particular the case of a binary
useful in calculating the phasing of GW signals from in- system containing a point particle (body 1) and an ex-
spiralling neutron star binaries. tendeddeformablestar(body2),workingtoquadrupolar
order in the star’s multipole series. We also discuss an
action principle formulation of the orbitaldynamics, the
E. Notation and Conventions processofenergytransferbetweenthe gravitationalfield
andthedeformablebody,andtheevolutionofthebody’s
We use units where Newton’s constant is G = 1, but spin due to tidal torques. Finally, we discuss the evolu-
retain factors of the speed of light c, with 1/c2 serv- tionofthe body’stidally inducedquadrupolemomentin
ing as the formal expansion parameter for the post- the adiabatic limit.
Newtonian expansion. We use lowercase latin letters
a,b,i,j,... for indices of(three-dimensional)spatialten-
sors. Spatial indices are contracted with the Eucliden A. Field equations
metric, viwi = δ viwj, with up or down placement of
ij
the indices having no meaning. We use uppercase latin In Newtonian physics, the scalar potential φ(t,x)
letters to denote multi-indices: L denotes the l indices obeys the Poissonequation,
a a ...a , K denotes the k indices b b ...b , etc. For a
1 2 l 1 2 k
given vector vi or for the partial derivative operator ∂ , 2φ=4πρ, (2.1)
i
∇
we use multi-indices or explicit sequences of indices to
denote their tensorial powers: with ρ(t,x) being the rest mass density of matter. The
influence ofthe gravitationalfieldonmatter isdescribed
vL = va1a2...al =va1va2...val, by the test particleaccelerationx¨i = ∂ φ, or moregen-
i
−
∂ = ∂ =∂ ∂ ...∂ , (1.15) erally, by Euler’s equation supplimented by the continu-
K b1b2...bk b1 b2 bk
ity equation (the conservation of mass),
and, for example, vij = vivj. We also use v2 = vii
danisdtin∇ct2t=ens∂oiris. ofMvualrtyi-iningdricaensk,areMa,lMsoau,Msedab,fo.r..se,twsitohf ∂t(ρvi)+∂j(ρvivj +tij)=−ρ∂iφ, (2.2)
ML = Ma1a2···al denoting the te{nsor of rank l. W}e use
the Einstein summation convention for both individual ρ˙+∂ (ρvi)=0, (2.3)
i
indices and multi-indices. Derivatives with respect to a
time coordinate t are denoted by ∂ or by overdots. with vi(t,x) being the matter’s velocity field, and
t
We use angular brackets to denote the symmetric, tij(t,x) the material stress tensor. Together, Eqs. (2.1-
trace-free (STF) projection of tensors [14]: 2.3) provide a complete description of Newtonian grav-
itational interactions. However, they do not in general
1
T<ab> = T(ab) δabTcc, form a closed set of evolution equations for the fields φ,
− 3 ρ,vi,andtij;oneneedsalsotospecifythematter’sinter-
3
T<abc> = T(abc) δ(abTc)dd, (1.16) nal dynamics, in particular concerning the stress tensor
− 5 t . (Inthe simplestcases,onecanfix tij byanalgebraic
ij
andsoon,withparenthesesdenotingthesymmetricpro- equation, like tij = 0 for ‘dust’ or tij = pδij with p(ρ)
jection. ForaSTFtensorSL =S<L> andgeneraltensor being the pressure for an isentropic perfect fluid.) Still,
TL, note that SLTL =SLT<L>. one can derive many useful results, like the form of the
translationalequationsofmotionforasystemofgravitat-
ing bodies, while leaving the matter’s internal dynamics
II. NEWTONIAN TIDAL INTERACTIONS unspecified.
In this section, we review the standard treatment of
tidal coupling in Newtonian theory [13]; the post-1- B. Multipole moments
Newtonian treatment given subsequent sections makes
extensiveuseoftheseNewtonian-orderresults. Wedefine We consider N isolated celestial bodies, i.e. regions of
themultipolemomentsandtidalmomentsofanextended space containing matter (ρ = 0) surrounded by regions
6
object and use them to derive the orbital (or transla- of vacuum (ρ = 0), and label the bodies by an index A,
tional) equations of motion for systems of gravitating with 1 A N. The potential that is locally generated
≤ ≤
7
by body A, whichwe will callthe internalpotential φint, about which the multipole expansion is centered to be
A
is given by the standard solution to (2.1) as an integral fixed to the body’s center of mass position:
over the volume of the body:
1
zi (t)= d3xρ(t,x)xi, (2.10)
ρ(t,x′) A M
φint(t,x)= d3x′ . (2.4) A Z
A − x x′
ZA | − | which is equivalent to (2.9b). Next comes the l = 2
Wecanexpressthispotentialasamultipoleseriesaround terminvolvingthebody’smassquadrupoletensorQij(t)
A
a (moving) point xi =zi (t) by using the Taylor series (2.9c), which we have renamed Mij Qij to accord
A A → A
with commonconvention. The higher-orderterms in the
1 ∞ ( 1)l 1 multipole series are supressed by increasing powers of
= − (x′ z )L∂ , (2.5)
x x′ l! − A L x z thefinitesizeparameterR/x¯ ,whereRisthesizeofthe
| − | Xl=0 | − A| body. | |
We note that the continuity equation(2.3) implies the
where L = a a ...a is a spatial multi-index, denoting
here tensorial1p2owerslof the vector (x′ z )i and of the constancy of the body’s total mass, M˙A =0. The Euler
A
− equation (2.2) similarly constrains the evolution of the
operator ∂ (c.f. Eq. (1.15)). Using the Taylor series
i
mass dipole Mi, thus determining the body’s transla-
(2.5)in(2.4)allowsustowritetheinternalpotential,for A
points xi exterior to the body, in the form tionalequationofmotion,asdiscussedinSec.IIDbelow.
Eqs. (2.2)and(2.3)donot,however,fully determinethe
∞ ( 1)l 1 evolutionofthequadrupoleandhighermultipoles,which
φiAnt(t,x)=− −l! MAL(t)∂L x z (t) , (2.6) will depend on the body’s internal dynamics.
A
l=0 | − |
X
with
C. Tidal moments
ML(t)= d3xρ(t,x)[x z (t)]<L>. (2.7)
A − A Having described the potential generated by an iso-
ZA lated body with its multipole moments, we can similarly
The quantities ML are the mass multipole moments of describe the potential felt by the body with tidal mo-
A
the body about the worldline zi (t). They are symmet- ments. Given a collection of several bodies indexed by
A
ric, trace-free spatial tensors of varying order l, ML = B,eachgivingriseits ownintrinsicpotentialofthe form
A
Ma1a2...al. The STF property follows from the fact that (2.6), we define the external potential felt by a given
A
∂ x−1 is an STF tensor, because partial derivatives body A to be the sum of the potentials due to all the
L
com| m|ute, and because ∂ x−1 = 2 x−1 = 0. As other bodies:
jj
themultipole momentsML|ar|econtra∇cte|d|withtheSTF
tensors ∂ x zA −1 in (2A.6), only their STF parts will φeAxt = φiBnt (2.11)
L
contribute|to−the p|otentialφint;thisiswhytheSTFpro- BX6=A
A
jection (denoted by angular brackets) has been included
The body’s tidal moments1 GL (t) are then defined as
in the definition of the multipole moments in (2.7). g,A
coefficients in the Taylor expansion of the external po-
The leading-order terms in the multipole series (2.6)
tential about the center-of-mass position zi :
can be written more explicitly as A
∞
1
φiAnt =−Mx¯A − 21QiAj∂ij x1¯ +... (2.8) φeAxt(t,x)=− l!GLg,A(t)[x−zA(t)]L, (2.12)
| | | | Xl=0
with
GL (t)= ∂ φext(t,x) . (2.13)
g,A − L A x=zA(t)
M = d3xρ, (2.9a)
A (cid:12)
ZA (cid:12)
0=Mi = d3xρx¯i, (2.9b)
A 1 The subscript g, standing for ‘global,’ has been included here
ZA
to avoid confusion with tidal moments introduced in our post-
1
Qij =Mij = d3xρ x¯ix¯j x¯ 2δij , (2.9c) Newtonian treatment below. We introduce there a set of body-
A A ZA (cid:18) − 3| | (cid:19) frame tidal moments GLA, defined in an accelerated reference
frameattached tothebody, andasetofglobal-frametidal mo-
agnenderx¯aite=dxbiy−thzeAit.otFailrmstaissstohfethmeobnoodpyoleMter(m2.9(al)=, a0n)d, tmideanltsmGomLg,Aen,tdsedfienfiendedininan(2(.a1s3s)ymcopintoctidicealwlyit)hintehretilaalttferarmate.NTewhe-
A
tonian order. While we have chosen to work exclusively in an
givingrisetoaCoulomb-typepotentialin(2.8). Wehave
inertial frame in our Newtonian treatment here, an analogous
omitted the dipole term (l=1) in (2.8) because Mi can
A Newtoniantreatmentthatusesaccelerated framescanbefound
always be made to vanish by choosing the point zi (t) inSec.IIIofRef.[15].
A
8
Like the multipole moments, the tidalmoments areSTF find the following EoMs:
tensors,GL =G<L>,ascanbeseenfromthedefinition
g,A g,A 1 1 1
(2.13) and the fact that 2φext = 0 everywhere outside M z¨i = M M ∂(1) + M Qjk∂(1)
∇ A 1 1 1 2 i z z 2 1 ijk z z
the bodies B = A. We see that G is simply (minus) 1 2 1 2
g,A | − | | − |
thepotentiala6tthebody’scenter,andGig,A isthewould- M z¨i = M M ∂(2) 1 + 1QjkM ∂(2) 1
be testparticleacceleration−∂iφeAxt. Forl≥2,theGLg,A 2 2 2 1 i |z2−z1| 2 1 ijk|z2−z1|
arehigher-orderderivativesofthepotentialthatwillgive
Defining the radius and unit vector associated with the
rise to tidal forces on a nonspherical body.
relative position,
The tidal moments of a body A can be expressed in
terms of the multipole moments ofthe other bodies B =
6 zi =zi zi, r = z , ni =zi/r, (2.17)
A by combining (2.6), (2.11), and (2.13), with the result 2− 1 | |
and using the general identity,
GL = ∂ φint(t,x)
g,A − L B x=zA(t)
B6=A 1 n<L>
X (cid:12) ∂ =( 1)l(2l 1)!! , (2.18)
∞ ( 1)k (cid:12) 1 Lr − − rl+1
= − MK∂(A) , (2.14)
k! B KL zA zB these EoMs can be written as
B6=Ak=0 | − |
X X
where ∂i(A) =∂/∂zAi , and ∂KL =∂b1...∂bk∂a1...∂al. M1z¨1i = −MMM2z¨2i 15M
= 1 2n + 1Qjkn<ijk> (2.19)
r2 i 2r4
D. Translational equations of motion
In this form, it is evident that the total momentum pi =
M z˙i+M z˙i isconserved,andthatthesetwoEoMsforzi
1 1 2 2 1
Aprimaryadvantageofthe languageofmultipole and andzi canbetradedfortheEoMoftherelativeposition
2
tidalmomentsisthatitallowsonetotakethePDEs(2.1- zi =zi zi:
2.3) governing the evolution of the fields ρ, vi, tij, and 2− 1
φ and extract from them ODEs for the center-of-mass M 15M
z¨i = ni Qjkn<ijk> (2.20)
worldlines zi (t) of a collection of gravitating bodies. To −r2 − 2M r4
A 2
this end, we consider a body A with multipole moments M 3M
MLdefinedby(2.7),inthepresenceofanexternalpoten- = ni Qjk(5nijk 2δijnk)
A −r2 − 2M r4 −
tial φext generated by other bodies B = A according to 2
A 6
(2.12) and (2.14). The body’s translational EoM can be where M = M +M is the total mass. In the second
1 2
found by applying two time derivatives to the definition line, we have used Eq. (1.16) to expand n<ijk> and the
of zi in (2.10), using the Euler and continuity equations fact that Qjk is STF.
A
(2.3) and (2.2), and intergrating by parts. The result is With Eq. (2.20), we have reduced the description of
an expressionfor the body’s center-of-massacceleration, the binary system’s translational dynamics to a single
ODE. Still, we can only solve this ODE if we know the
M z¨i = d3xρ∂ φext, (2.15) time evolution of the quadrupole moment Qij(t), which
A A − i A requires a detailed model of body 2’s interior. We will
Z
describe a simple adiabatic model for Qij in Sec. IIG
which can be rewritten in terms of the body’s multipole
below.
and tidal moments by using (2.12) and (2.7):
∞
M z¨i = 1MLGiL (2.16) E. Action principle
A A l! A g,A
l=0
X
1 As for any Newtonian system, a Lagrangian for a col-
= MAGig,A+ 2QjAkGigj,kA+... lection of gravitating bodies can be constructed from
=T U,withT beingthekineticenergyandU thepo-
L −
Thefirstterminthesecondlinerepresentstheforcethat tential energy. The total kinetic energy receves separate
would act on a freely falling test mass M , while the contributions from each body, T = T , and each T
A A A A
second term gives the leading-order tidal force. can be split into a contribution from the center-of-mass
P
TorendertheEoM(2.16)fullyexplicit,onecanusethe motion of the body and an internal contribution:
expressions for the tidal moments (2.14) in terms of the
1 1
multipole moments and worldlines of the other bodies. T = d3xρv2 = M z˙2 +Tint, (2.21)
Consideringthecaseofatwo-bodysystemA=1,2,with A 2ZA 2 A A A
body1havingonlyamonopolemomentM ,andbody 2 1
having a monopole M2 and a quadrupole Q1ij ≡ Qi2j, we TAint = 2ZAd3xρ(v−z˙A)2. (2.22)
9
Here,wehaveusedthefactthat d3xρvi =M z˙i ,im- (We have omitted the internal Lagrangian for body 1
A A A
plied by the continuity equation (2.3) and the definition as it is completely decoupled from the rest of the sys-
of the center-of-mass position zRi in (2.10). The gravi- tem.) Varying this action with respect to zi and zi
A 1 2
tational potential energy U = U can be similarly leadstotheir EoMsfoundin(2.19). Alternately,varying
A A
split into external and internal parts: with respect to their separation zi = zi zi gives the
P 2 − 1
relative EoM (2.20), while the system’s center of mass
1
UA = 2ZAd3xρφ=UAext+UAint, (2.23) t(Mhe1Lz1aig+raMng2iaz2ni)(/2M.27i)stfoouthnedcetontebre-ocf-ymclaics.sfrSapmecei,awlihzienrge
1 M zi +M zi =0, gives
Uint = d3xρφint, (2.24) 1 1 2 2
A 2 A
ZA µz˙2 µM
Uext = 1 d3xρφext = 1 ∞ 1MLGL . (2.25) L= 2 + r −UQ+Li2nt, (2.29)
A 2 A −2 l! A g,A
ZA l=0
X with µ=M M /M being the reduced mass.
1 2
In the last line, we have used (2.12) to express φext in While leaving the functions int(qα,q˙α) and Qij(qα)
A L2 2 2 2
termsofthetidalmomentsand(2.7)forthedefinitionof unspecified, we can still use the Lagrangian (2.27) to
the multipole moments. writedownEoMsforbody2’sinternalconfigurationvari-
While the system’s total potential energy will also re- ables q2α:
cieve non-gravitational contributions from the internal
d ∂ int ∂ int 1 ∂Qij
structureofeachbody, wecanlumpthese contributions, L2 = L2 + Gij . (2.30)
along with Tint and Uint as defined above, into an inter- dt ∂q˙α ∂qα 2 g,2 ∂qα
A A 2 2 2
nal Lagrangian int for each body. We can then write
LA which will be useful in the next subsection.
the total Lagrangianfor an N-body system as
∞
1 1 1
= M z˙2 + MLGL + int . (2.26) F. Energy
L 2 A A 2 l! A g,A LA !
A l=0
X X
Continuing to specialize to the two-body M -M -Q
Thebodies’center-of-massworldlineszi (t)enterthisLa- 1 2 2
A case,wecanconstructaconservedenergyforthesystem
grangian through the translational kinetic energy terms
from
and through the external gravitational potential energy
terms (via the tidal moments); the internal Lagrangians ∂ ∂ ∂
LcoiAnnts,trhuocwtieovne.r,Tahreeindinetpwenildlebnet foufntchteiownsoroldflsinomesezsAie,tboyf E =T +U = ∂zL˙1iz˙1i + ∂zL˙2iz˙2i + ∂qL˙2αq˙2α−L, (2.31)
internalconfiguratioLnAvariablesqα (andtheirtimederiva- with summationoverα implied. Using the (CoM-frame)
A
tives) for each body, which will include e.g. Euler angles Lagrangian(2.29), we find
for the orientation of the body, vibrational mode ampli-
tudes,etc.,dependingonthemodelofthebody’sinternal µz˙2 µM
E = +U +Eint, (2.32)
structure. (Infullgenerality,theproperinternalconfigu- 2 − r Q 2
rationvariables qα are the fields ρ, vi and tij, subject to
A
where U is given by (2.28), and the internal energy of
(2.10) as a constraint.) The bodies’ multipole moments Q
ML, for l 2, appearing in the gravitational potential body 2 is given by
A ≥
energy terms in (2.26), will be functions of these same
∂ int
internal variables qα. Together, the zi and qα, for all Eint(qα,q˙α)= L2 q˙α int (2.33)
bodies A, form a coAmplete set of dynamAical varAiables for 2 2 2 ∂q˙2α 2 −L2
the N-body system. Varying the action S = dt with
respectto the worldlineszi reproducesthe tranLslational Thisinternalenergywillgenerallyhaveseveralcontribu-
A R tions: internal gravitational potential energy, rotational
EoMs (2.16). Determining the evolution of the variables
kinetic energy, vibrational kinetic and potential energy,
qα, and hence the moments ML for l 2, will require a
mAodel for int(qα,q˙α) and MLA(qα). ≥ thermal energy, etc. Nonetheless, the rate at which en-
LA A A A A ergy is exchanged between the interior of body 2 and its
Specializingtothetwo-bodyM -M -Q caseandusing
1 2 2 surroundings, via the gravitational tidal interaction, is
(2.14),(2.17),and(2.18),theLagrangian(2.26)becomes
a function only of the orbital separation, M , and Qij.
1
1 1 M M Differentiating (2.33) with respect to time and using the
L= 2M1z˙12+ 2M2z˙22+ 1r 2 −UQ+Li2nt. (2.27) EoM (2.30) for the internal variables q2α, we find
where UQ is the potentialenergy ofthe quadrupole-tidal E˙int = q˙α d ∂Li2nt ∂Li2nt = 1Gij ∂Qijq˙α
interaction: 2 2 dt ∂q˙α − ∂qα 2 g,2 ∂qα 2
(cid:18) 2 2 (cid:19) 2
1 3M 1 3M
U = QijGij , Gij = 1n<ij>. (2.28) = Gij Q˙ij = 1nijQ˙ij (2.34)
Q −2 g,2 g,2 r3 2 g,2 2r3
10
The energy transfer described by (2.34) is often referred to linear order in the tidal deformability λ.
to as tidalheating (see e.g.[25]). This expressionfor the The internal energy Eint (2.33), in the adiabatic ap-
2
power delivered to the body is valid (in the quadrupolar proximation, is given by
approximation)regardlessof the body’s internaldynam-
1
ics. Using (2.34) and the orbital EoM (2.20), one can Eint = QabQab, (2.40)
2 4λ
confirm that the binary system’s total energy (2.32) is
conserved. up to a constant,and satisfies the tidalheating equation
(2.34)byvirtueofEq.(2.35). (Weshouldnotethatthere
is actually no ‘heating’ taking place here, as this model
G. Adiabatic approximation neglects dissapative effects and is completely conserva-
tive.) Using Eq. (2.35), the binary system’s total energy
While we have thus far left unspecified the internal E (2.32) can be written as
dynamics for the deformable body 2, which determines
the evolution of the quadrupole, we now specialize our µz˙2 µM 3λM1
E = 1+ , (2.41)
analysistothecasewhereQij(t)isadiabaticallyinduced 2 − r 2r5M
(cid:18) 2(cid:19)
by the tidal field. This will lead to a closed system of
in the adiabatic model, and is conserved by the orbital
evolutionequationsforthebinary. Intheadiabaticlimit,
EoM (2.38). Then, using z˙2 = r2ω2 and Eq. (2.39), we
whenthebody’sinternaldynamicaltimescalesaremuch
find
less than the orbital period, the qudrupole will respond
to the instaneous tidal field according to µ 9λM ω10/3
E(ω)= (Mω)2/3 1 1 , (2.42)
Qij(t)=λGigj,2(t), (2.35) −2 (cid:18) − M2M5/3 (cid:19)
for the circular-orbitenergy-frequency relationship.
whereλisthetidaldeformability,andGab (t)isthetidal
g,2
tensorgiveninEq.(2.28). AsdiscussedinSec.IBandin
more detail in Refs. [8, 9], the relation (2.35) should be
H. Spin
validto 1%forneutronstarbinariesatGWfrequencies
∼
.400 Hz.
In anticipation of the 1PN treatment of the binary’s
Theadiabaticevolutionofthequadrupolecanbeinco-
proratedintoouractionprinciple(2.29)bytakingQij(t) orbital dynamics, in which a body’s angular momentum
tobeourloneinternalconfigurationvariable(qα),andby (or spin) has a direct influence on the orbit, it will be
taking the internal Lagrangian int to contain2 a simple usefultodiscusstheevolutionofanextendedbody’sspin
L2 atNewtonianorder. ThespinofabodyAaboutitsCoM
quadratic potential energy cost for the quadrupole:
worldline zi is defined by
A
µz˙2 µM
[zi,Qij] = + U + int (2.36)
L 2 r − Q L2 Sa(t)=ǫabc d3xρ(t,x) xb zb(t) vc(t,x), (2.43)
= µz˙2 + µM + 1Gab Qab 1 QabQab. A Z (cid:2) − A (cid:3)
2 r 2 g,2 − 4λ with ρ being the mass density and vc the velocity field.
Takingatimederivativeofthisequation,usingtheEuler
Varyingthe actionwithrespectto the orbitalseparation
zi still gives the orbital EoM (2.20), and varying with and continuity equations (2.2) and (2.3) and the defini-
respect to the quadrupole Qij reproduces the adiabatic tion of zAi in (2.10), and integrating by parts, we find
evolutionequation(2.35). UsingEq.(2.35)toreplaceQij
and Eq. (2.28) for Gag,b2, the Lagrangian can can written S˙Aa = −ǫabc d3xρ(xb−zAb)∂cφeAxt
solely in terms of zi as Z
∞
1
[zi]= µz˙2 + µM 1+ 3λM1 . (2.37) = ǫabc l!MAbLGcgL,A. (2.44)
L 2 r 2r5M Xl=0
(cid:18) 2(cid:19)
Inthesecondline,wehaveusedthedefinitionsofMLand
ThisLagrangianleadstotheorbitalEoM(2.20)withQij A
GL in (2.7) and (2.13). This formula gives the torque
replaced by its adiabatic value (2.35), g,A
on the body due to tidal forces. As it is not directly
Mni 9λM relevanttoourpurposes,wewillnotdiscussaLagrangian
ai = 1+ 1 , (2.38)
− r2 r5M formulation of the Newtonian rotational dynamics.
(cid:18) 2(cid:19) Applying Eq.(2.44) to the M -M -Q system, we find
1 2 2
which shows that the tidal coupling results in an attrac- that the tidal torque on body 2 is given by
tive force. For circularorbits,withai = rω2ni, we find
the radius-frequency relationship − S˙2a =ǫabcQbdGcgd,2 (2.45)
r(ω)= M1/3 1 3λM1ω10/3 , (2.39) with the tidal tensor Gcgd,2 given by (2.28). Eq. (2.45) is
ω2/3 − M M5/3 valid (in the quadrupolar approximation) regardless of
(cid:18) 2 (cid:19)